Using Chebyshev's theorem, the solution to the problems posed are given thus :
Since the distribution isn't normally distributed, we use Chebyshev's theorem :
- [1 - (1/k²)]
- k = number of standard deviations from the mean.
- Mean = 38
- Standard deviation = 6
1.)
Proportion of values that fall between 26 and 50 :
(26 - 38) ; (50 - 38) = - 12, 12 = 12/6 = 2 standard deviations from the mean :
Hence, we have ;
[1 - (1/2²)]
[1 - 1/4] = 3/4 = 0.75
Hence, about 75% of the values fall between 26 and 50
2.)
Proportion that fall between 14 and 62 :
(62 - 38) / 6 = 24/6 = 4 standard deviations from the mean
Hence, we have ;
[1 - (1/4²)]
[1 - 1/16] = 15/16 = 0.9375
Hence, about 93.75% of the values fall between 14 and 62
3.)
We solve for k in the equation :
89% = 0.89
0.89 = [1 - 1/k²]
0.89 = (k² - 1)/k²
0.89k² = k² - 1
0.89k² - k² = - 1
0.11k² = - 1
k² = - 1/-0.11
k² = 9. 09
k = √9.09
k = 3.01 = 3
Lower boundary = 38 - (3×6) = 38 - 18 = 20
Upper boundary = 38 + (3×6) = 38 + 18 = 56
Hence, the two values are (20, 56)
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